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S3 Maths | Factorising Quadratics Lesson Prep Guide for Tutors | Three Methods and Cross Method
Students can expand (x+3)(x+4) but guess wildly in reverse. Factorising is expansion played backwards, and expansion is always available as a check.
Objectives — Students can: (1) take out the common factor before choosing a method; (2) use the cross method with leading coefficient one and not one; (3) expand to verify every answer, scoring at least 7 of 10 in the diagnostic.
01 Prerequisites (3-minute check)
If any of these fails, fix it before the lesson starts.
| Skill | Quick check | Pass mark |
|---|---|---|
| Expanding brackets | Expand (x+3)(x+4) | x squared plus 7x plus 12 |
| Multiplying negatives | What is -2 times -3? | 6 |
| Highest common factor | Common factor of 6x and 12 | 6 |
02 Lesson flow (60 minutes)
| Time | Stage | Tutor does | Student does |
|---|---|---|---|
| 0-6 min | Play it backwards | Expand (x+3)(x+4), then ask which brackets give x squared plus 7x plus 12 | Watch and answer |
| 6-18 min | Common factor first | Model 3x squared minus 12x as 3x(x-4) | Factorise three expressions |
| 18-32 min | Cross method, coefficient one | Two numbers that add to the middle and multiply to the constant | Do four and name the pair |
| 32-46 min | Coefficient not one and squares | Model 2x squared plus 7x plus 3, 6x squared minus 7x minus 3, and x squared minus 9 | Two of each |
| 46-56 min | 10-question diagnostic | Mixed types, marked immediately: 7 right and 3 wrong | Work independently |
| 56-60 min | Exit check | Three oral questions, then reteach the weak method | Answer orally |
03 Teaching script (read it out)
Scene 1: Always take the common factor first
Ask one question before anything else: do the terms share a factor? For 3x squared minus 12x the answer is 3x, so the expression becomes 3x(x-4), and only then does the bracket deserve attention. Students often start writing brackets immediately and stall; taking the factor out is zero risk and usually simplifies the question.
Student task: factorise 3x squared minus 12x, 2x squared plus 6x and 5x squared minus 20 by taking the common factor.
Scene 2: Cross method: two numbers that add and multiply
With leading coefficient one, x squared plus 7x plus 12 needs two numbers adding to 7 and multiplying to 12: 3 and 4, so the answer is (x+3)(x+4). When the constant is positive and the middle negative, both numbers are negative: x squared minus 5x plus 6 gives -2 and -3. Difference of squares is the special case: x squared minus 9 is (x+3)(x-3).
Student task: say the sum and the product out loud before writing the brackets.
Scene 3: Coefficient not one, then verify by expanding
For 2x squared plus 7x plus 3 use 2x and x, then pair 1 with 3: the cross terms give 6x plus x, which is 7x, so the answer is (2x+1)(x+3). For 6x squared minus 7x minus 3 use 3x and 2x with 1 and -3: the cross terms give -9x plus 2x, which is -7x. Always expand at the end; if the expansion does not match the question, a sign slipped somewhere.
Student task: expand (2x+1)(x+3) and (3x+1)(2x-3) to confirm they return to the originals.
04 Question types and worked steps
| Type | Example | Steps | Answer |
|---|---|---|---|
| Common factor | 3x squared minus 12x | Take out 3x, leaving x-4 | 3x(x-4) |
| Cross method, coefficient one | x squared plus 7x plus 12 | 3 and 4: sum 7, product 12 | (x+3)(x+4) |
| Difference of squares | x squared minus 9 | Read as x squared minus 3 squared | (x+3)(x-3) |
| Coefficient not one | 6x squared minus 7x minus 3 | 3x and 2x with 1 and -3: -9x plus 2x | (3x+1)(2x-3) |
Sanity check (teach this): Sanity checks: every answer expands back to the question; x squared minus 9 is not (x-3) squared; x squared plus 9 does not factorise over the reals; the cross terms must add to -7x; 7 of 10 is 70 percent, not 7 percent.
05 Independent practice (3 questions)
| Question | Answer | What to watch |
|---|---|---|
| Three common factors | 3x squared minus 12x; 2x squared plus 6x; 5x squared minus 20 | Factor taken out completely |
| Four cross-method cases | x squared plus 7x plus 12; minus 5x plus 6; plus 4x plus 3; minus 9 | Sum and product of the pair |
| Two harder cases | 2x squared plus 7x plus 3; 6x squared minus 7x minus 3 | Cross terms give the middle |
06 Common mistakes and what to say
| Mistake | Student writes | Say this |
|---|---|---|
| Missing common factor | Leaves 3x outside only halfway through | Take 3x out first and write 3x(x-4), not a half-finished bracket. |
| Wrong signs | Writes (x+2)(x+3) for x squared minus 5x plus 6 | Middle negative and constant positive means both numbers are negative. |
| Squares as a perfect square | Writes (x-3) squared for x squared minus 9 | (x-3) squared has a -6x term, so the answer is (x+3)(x-3). |
| Cross terms reversed | Writes (3x-1)(2x+3) | The cross terms must give -7x, not +7x. |
07 Homework (paper, section and question numbers)
- Secondary maths chapter index — All chapters
- Master prep index — Compare related chapters
- Redo this chapter — Redo all 10 questions and expand every answer
08 Exit check (last 5 minutes)
| Question | Answer | Decision |
|---|---|---|
| Factorise x squared plus 7x plus 12 | (x+3)(x+4) | Cannot do it: reteach act two |
| Factorise x squared minus 9 | (x+3)(x-3) | (x-3) squared: reteach act three |
| Factorise 6x squared minus 7x minus 3 | (3x+1)(2x-3) | Sign error: reteach act three |
09 Tutor FAQ
Must students memorise a cross table?
No. Drill the sum and product pairing first, then cross terms when the leading coefficient is not one.
When is difference of squares allowed?
Two terms, both perfect squares, and a minus sign between them. A plus sign means it does not factorise over the reals.
Why expand every answer?
Expansion is the only check that does not rely on memory; it takes ten seconds and it is worth doing under exam pressure.
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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.